The Swan can only hold 4, but with double beds, the Dolphin can have a rollaway bed, so you can have 5. It will be a bit tight, though, so it's a compromise: stay in the best area (in my opinion) of Disney, but be kind of cramped into one room.
Oops, I just checked and it looks like the Dolphin is booked up for the 19th to the 23rd.
The only decent alternative I can think of it to check out the Downtown Disney hotels. They all have regular transportation to the theme parks, if it's maybe not quite as frequent as the Disney resorts, still it's there and included in your room rate. For 5 people, I'd look into the Doubletree Guest Suites, as you'd have a bedroom with 2 beds, plus a sofa sleeper (years ago, I slept on it while visiting with my parents, and it wasn't bad) plus you'd have a fridge and microwave. The new Holiday Inn looks nice too.
The Hilton Downtown Disney gets EMH.
The Sheraton Safari's website says they have shuttles, and they have suites listed for about $134 a night for your dates.
I'm not sure if the Hilton Bonnet Creek has shuttles, but I think they do.
Good luck!